I read the Wiki, but I am still somewhat confused. Supposedly floor returns the number lowest, and ceil the number highest. However, I am not sure I understand how this works in freehand form, or how I may implement this myself in a specific thing. Would someone please clarify how ceil and floor works, and how it is used/examples? Thanks!!
$\endgroup$ 133 Answers
$\begingroup$Simply put it's two ways of thinking of rounding off a number. Ceiling rounds up to nearest integer. Floor rounds down to nearest integer. If the number is an integer, nothing happens.
$\endgroup$ $\begingroup$It's easy to think about floor and ceil from the perspective of the number line.
Let's say you have some decimal number, $2.31$ (I'm going to be using this number as an example throughout my answer)$\hskip2in$
So, as you can see, the functions just return the nearest integer values.
floor returns the nearest lowest integer and ceil returns the nearest highest integer.
All real numbers are made of a characteristic (an integer part) and mantissa (a fractional part) $$\text{Number} = \text{Characteristic} + \text{Mantissa}$$ $$2.31 = 2 + 0.31$$
When floor a number, you can think of it as replacing the Mantissa with $0$ $$\lfloor 2.31 \rfloor = 2 + 0 = 2$$
and ceil can be thought of as replacing the mantissa with $1$. $$\lceil 2.31 \rceil = 2 + 1 = 3$$
That's not a very popular way of thinking about it but it was the way I thought about it when I first started using it in programming.
Remember, the number remains the same when it is an integer. ie, floor($3$) $=$ ceil($3$) $= 3$
Let's now look at the proper definitions along with the graphs for them.
Floor Function: Returns the greatest integer that is less than or equal to $x$
$\hskip2in$
Ceiling Function: Returns the least integer that is greater than or equal to $x$
$\hskip2in$
Don't let the infinite staircase scare you. It's much more simpler than it seems. Those "line-segments" that you see are actually called piecewise-step functions.
Simply, the black dot represents 'including this number' and the white represents 'excluding this number'. Meaning that each segment actually is from x to all numbers less than x+1.
Let's look at 2.31 and how it would look on both the graphs at once.
Here's an example of using the floor function: I have a number $m$ and I am looking for the smallest number $\ell$ such that $2^\ell\gt m+1$. To find such a number I would use the expression
$$\ell =[\log_2(m+1)]+1$$
Where the square brackets indicate the use of the floor function.
The floor and ceil functions in a programming language might use a combination of truncation (rounding towards zero) and comparisons to actually achieve their result. For example, if int is a function that truncates a number to its integer part, then for $x\ge0$ we have floor(x)=int(x), but for $x\lt 0$ it would be
y=int(x)
if (y\ne x) floor(x)=y-1
else floor(x)=y
